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Theorem canthwdom 9032
Description: Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 8659, equivalent to canth 7103). (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
canthwdom ¬ 𝒫 𝐴* 𝐴

Proof of Theorem canthwdom
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 5253 . . . . 5 ∅ ∈ 𝒫 𝐴
2 ne0i 4304 . . . . 5 (∅ ∈ 𝒫 𝐴 → 𝒫 𝐴 ≠ ∅)
31, 2mp1i 13 . . . 4 (𝒫 𝐴* 𝐴 → 𝒫 𝐴 ≠ ∅)
4 brwdomn0 9022 . . . 4 (𝒫 𝐴 ≠ ∅ → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
53, 4syl 17 . . 3 (𝒫 𝐴* 𝐴 → (𝒫 𝐴* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴))
65ibi 268 . 2 (𝒫 𝐴* 𝐴 → ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
7 relwdom 9019 . . . . 5 Rel ≼*
87brrelex2i 5608 . . . 4 (𝒫 𝐴* 𝐴𝐴 ∈ V)
9 foeq2 6584 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝑥))
10 pweq 4545 . . . . . . . 8 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
11 foeq3 6585 . . . . . . . 8 (𝒫 𝑥 = 𝒫 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1210, 11syl 17 . . . . . . 7 (𝑥 = 𝐴 → (𝑓:𝐴onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
139, 12bitrd 280 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑥onto→𝒫 𝑥𝑓:𝐴onto→𝒫 𝐴))
1413notbid 319 . . . . 5 (𝑥 = 𝐴 → (¬ 𝑓:𝑥onto→𝒫 𝑥 ↔ ¬ 𝑓:𝐴onto→𝒫 𝐴))
15 vex 3503 . . . . . 6 𝑥 ∈ V
1615canth 7103 . . . . 5 ¬ 𝑓:𝑥onto→𝒫 𝑥
1714, 16vtoclg 3573 . . . 4 (𝐴 ∈ V → ¬ 𝑓:𝐴onto→𝒫 𝐴)
188, 17syl 17 . . 3 (𝒫 𝐴* 𝐴 → ¬ 𝑓:𝐴onto→𝒫 𝐴)
1918nexdv 1930 . 2 (𝒫 𝐴* 𝐴 → ¬ ∃𝑓 𝑓:𝐴onto→𝒫 𝐴)
206, 19pm2.65i 195 1 ¬ 𝒫 𝐴* 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1530  wex 1773  wcel 2107  wne 3021  Vcvv 3500  c0 4295  𝒫 cpw 4542   class class class wbr 5063  ontowfo 6350  * cwdom 9010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fo 6358  df-fv 6360  df-wdom 9012
This theorem is referenced by:  pwdjudom  9627
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